Whiteboard

Course Goals

Discussion on Syllabus.

Why learn the theory of computation?

• Dijkstra suggests that theory is absolutely necessary while Paul Graham suggests that theory is practically useless. Possibly the truth lies in between.
• Addition is easier than multiplication. How? Why?

Ingredients for computation:

• Data as representations of objects
• Algorithms as operations on data

The way one represents data affects how one computes with it.

For example, take Roman Numerals vs the Place Value System.

The distance to the moon is 238,000 miles—using the Roman system would take multiple pages to represent this number.

This also applies to algorithms as well.

For example, take multiplication.

Algorithms where $\log i$ is # digits and $i_1,\,i_2$ are numbers:

• Repeated Addition: $\mathcal{O}(i)$
• Classic (Grade School): $\mathcal{O}((\log i)^2)$

Clearly, a bad algorithm can make or break computation.

A good algorithm on a weaker computer >>> a bad algorithm on a modern computer

The best algorithm for multiplication is $\mathcal{O}(n\log n)$

What can/can’t you actually compute? (Can a computer solve problem $P$?)

• What does it mean to solve a problem?
• How can we model computation?
• How can we define what an algorithm is?
• Turing model?